Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic's current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.

Just three weeks later -- a blink of an eye compared to the usual pace of mathematics journals -- Zhang received the referee report on his paper.

"The main results are of the first rank," one of the referees wrote. The author had proved "a landmark theorem in the distribution of prime numbers."

Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know -- someone whose talents had been so overlooked after he earned his doctorate in 1992 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.

"Basically, no one knows him," said Andrew Granville, a number theorist at the Universite de Montreal. "Now, suddenly, he has proved one of the great results in the history of number theory."

Reminds me of a certain patent clerk and his theories about time and space. History doesn't repeat itself, but it does rhyme. (via @daveg)

Erica Klarreich, a Berkeley-based science writer who has a Ph.D. in mathematics and has written about Zhang, says his proof demonstrates the remarkable balance between order and randomness within the prime numbers. "Prime numbers are anything but random -- they are completely determined," Klarreich says. "Nevertheless, they seem to behave in many respects like randomly-sprinkled numbers that eventually display all possible clumps and clusters. Zhang's work helps to put this conjectured picture of the primes on a solid footing."

Update: Alec Wilkinson has a profile of Zhang in the Feb 2, 2015 issue of the New Yorker: The Pursuit of Beauty.

Zhang, who also calls himself Tom, had published only one paper, to quiet acclaim, in 2001. In 2010, he was fifty-five. "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game," Hardy wrote. He also wrote, "I do not know of an instance of a major mathematical advance initiated by a man past fifty." Zhang had received a Ph.D. in algebraic geometry from Purdue in 1991. His adviser, T. T. Moh, with whom he parted unhappily, recently wrote a description on his Web site of Zhang as a graduate student: "When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the North Pole." Zhang left Purdue without Moh's support, and, having published no papers, was unable to find an academic job. He lived, sometimes with friends, in Lexington, Kentucky, where he had occasional work, and in New York City, where he also had friends and occasional work. In Kentucky, he became involved with a group interested in Chinese democracy. Its slogan was "Freedom, Democracy, Rule of Law, and Pluralism." A member of the group, a chemist in a lab, opened a Subway franchise as a means of raising money. "Since Tom was a genius at numbers," another member of the group told me, "he was invited to help him." Zhang kept the books. "Sometimes, if it was busy at the store, I helped with the cash register," Zhang told me recently. "Even I knew how to make the sandwiches, but I didn't do it so much." When Zhang wasn't working, he would go to the library at the University of Kentucky and read journals in algebraic geometry and number theory. "For years, I didn't really keep up my dream in mathematics," he said.

It's a bummer that Alec Wilkinson's article on free diving isn't available online (except for NYer subscribers)...it's fascinating and right up the alley of the relaxed concentration/deliberate practice enthusiast. One of the two divers profiled uses a technique called attention deconcentration to govern her body and mind as she dives.

To still the unbidden apprehensions that might interfere with her dive -- what she describes as "the subjective feeling of empty lungs at the deep" -- Molchanova uses a technique that she refers to as "attention deconcentration." ("They get it from the military," Ericson said.) Molchanova told me, "It means distribution of the whole field of attention -- you try to feel everything simultaneously. This condition creates an empty consciousness, so the bad thoughts don't exist."

"Is it difficult to learn?"

"Yes, it's difficult. I teach it in my university. It's a technique from ancient warriors -- it was used by samurai -- but it was developed by a Russian scientist, Oleg Bakhtiyarov, as a psychological-state-management technique for people sho do very monotonous jobs."

I asked if it was like meditation.

"To some degree, except meditation means you're completely free, but if you're in the sea at depth you will have to be focussed, or it will get bad. What you do to start learning is you focus on the edges, not the center of things, as if you were looking at a screen. Basically, all the time I am diving, I have an empty consciousness. I have a kind of melody going through my mind that keeps me going, but otherwise I am completely not in my mind."

Rising from the depth, it is important to constantly scan your condition to prevent shallow water black-out, which can occur without any discomfort sensations. Somatic attention deconcentration appears to be extremely useful in this situation. Somatic AD implies attention distribution on the whole volume of the body and allows noticing tiny changes of organism state.

There is one more kind of AD -- aural attention deconcentration. It is not so effective in the water, but it helps preparing to the dive and not to be distracted by judge's countdown.

It's interesting that both the attention deconcentration and flow techniques are designed to get the practitioner to basically the same place (i.e. ready to perform difficult tasks) from opposite directions.

Somewhat related, a reader (thx, martin) recently sent in a link to The Game, a mind game with an unusual objective:

The Game is an ongoing mind game, the objective of which is to avoid thinking about The Game itself. Thinking about The Game constitutes a loss, which, according to the rules of The Game, must then be announced. How to win The Game is not defined in the rules; players can only attempt to avoid losing for as long as possible. The Game has been described alternately as pointless and infuriating, or as a challenging game that is fun to play.